* Intervals are closed, connected subsets of the real numbers. Intervals may be unbound (in either or both directions) or empty. In special cases <code>+inf</code> and <code>-inf</code> are used to denote boundaries of unbound intervals, but any member of the interval is a finite real number.

+

* Classical functions are extended to interval functions as follows: The result of function f evaluated on interval x is an interval '''enclosure of all possible values''' of f over x where the function is defined. Most interval arithmetic functions in this package manage to produce a very accurate such enclosure.

+

* The result of an interval arithmetic function is an interval in general. It might happen, that the mathematical range of a function consist of several intervals, but their union will be returned, e. g., 1 / [-1, 1] = [Entire].

−

Warning: The package has not yet been released.

+

__TOC__

+

[[File:Interval-sombrero.png|280px|thumb|left|Example: Plotting the interval enclosure of a function]]

{{quote|Give a digital computer a problem in arithmetic, and it will grind away methodically, tirelessly, at gigahertz speed, until ultimately it produces the wrong answer. … An interval computation yields a pair of numbers, an upper and a lower bound, which are guaranteed to enclose the exact answer. Maybe you still don’t know the truth, but at least you know how much you don’t know.|Brian Hayes|[http://dx.doi.org/10.1511/2003.6.484 DOI: 10.1511/2003.6.484]}}

+

== Development status ==

+

* Completeness

+

** All required functions from [https://standards.ieee.org/findstds/standard/1788-2015.html IEEE Std 1788-2015], IEEE standard for interval arithmetic, are implemented. The standard was approved by IEEE-SA on June 11, 2015. It will remain active for ten years. The standard was approved by ANSI in 2016.

+

** Also, the minimalistic standard [https://standards.ieee.org/findstds/standard/1788.1-2017.html IEEE Std 1788.1-2017], IEEE standard for interval arithmetic (simplified) is fully implemented. The standard was approved by IEEE-SA on December 6, 2017 (and published in January 2018).

+

** In addition there are functions for interval matrix arithmetic, N-dimensional interval arrays, plotting, and solvers.

** Includes [https://github.com/oheim/ITF1788 large test suite] for arithmetic functions

+

** For open bugs please refer to the [https://savannah.gnu.org/search/?words=forge+interval&type_of_search=bugs&only_group_id=1925&exact=1 bug tracker].

+

* Performance

+

** All elementary functions have been [https://octave.org/doc/interpreter/Vectorization-and-Faster-Code-Execution.html vectorized] and run fast on large input data.

+

** Arithmetic is performed with the [http://www.mpfr.org/ GNU MPFR] library internally. Where possible, the optimized [http://web.archive.org/web/20170128033523/http://lipforge.ens-lyon.fr/www/crlibm/ CRlibm] library is used.

+

* Portability

+

** Runs in GNU Octave ≥ 3.8.2

+

** Known to run under GNU/Linux, Microsoft Windows, macOS, and FreeBSD

−

{| class="wikitable" style="margin: auto"

+

== Project ideas (TODOs) ==

−

!Standard floating point arithmetic

+

* To be considered in the future: Algorithms can be migrated from the C-XSC Toolbox (C++ code) from [http://www2.math.uni-wuppertal.de/wrswt/xsc/cxsc_new.html] (nlinsys.cpp and cpzero.cpp), however these would need gradient arithmetic and complex arithmetic.

d) It should be possible to output the bounds of an interval without punctuation,

+

e.g., 1.234 2.345 instead of [1.234, 2.345]. For instance, this might be a

+

convenient way to write intervals to a file for use by another application.

−

Floating point arithmetic, as specified by [http://en.wikipedia.org/wiki/IEEE_floating_point IEEE 754], is available in almost every computer system today. It is wide-spread, implemented in common hardware and integral part in programming languages. For example, the extended precision format is the default numeric data type in GNU Octave. Benefits are obvious: The performance of arithmetic operations is well-defined, highly efficient and results are comparable between different systems.

+

== Compatibility ==

+

The interval package's main goal is to be compliant with IEEE Std 1788-2015, so it is compatible with other standard-conforming implementations (on the set of operations described by the standard document). Other implementations, which are known to aim for standard conformance are:

−

However, there are some downsides of floating point arithmetic in practice, which will eventually produce errors in computations.

* Even if the developer would be proficient, most developing environments / technologies limit floating point arithmetic capabilities to a very limited subset of IEEE 754: Only one or two data types, no rounding modes, …

−

* Results are hardly predictable. All operations produce the best possible accuracy ''at runtime'', this is how floating point works. Contrariwise, financial computer systems typically use a [http://en.wikipedia.org/wiki/Fixed-point_arithmetic fixed-point arithmetic] (COBOL, PL/I, …), where overflow and rounding can be precisely predicted ''at compile-time''.

−

* If you do not know the technical details, cf. first bullet, you ignore the fact that the computer lies to you in many situations. For example, when looking at numerical output and the computer says “<code>ans = 0.1</code>,” this is not absolutely correct. In fact, the value is only ''close enough'' to the value 0.1.

−

Interval arithmetic addresses above problems in its very special way and introduces new possibilities for algorithms. For example, the [http://en.wikipedia.org/wiki/Interval_arithmetic#Interval_Newton_method interval newton method] is able to find ''all'' zeros of a particular function.

+

=== Octave Forge simp package ===

+

In 2008/2009 there was a Single Interval Mathematics Package (SIMP) for Octave, which has eventually become unmaintained at Octave Forge.

−

== Theory ==

+

The simp package contains a few basic interval arithmetic operations on scalar or vector intervals. It does not consider inaccurate built-in arithmetic functions, round-off, conversion and representational errors. As a result its syntax is very easy, but the arithmetic fails to produce guaranteed enclosures.

−

=== Moore's fundamental theroem of interval arithmetic ===

+

It is recommended to use the interval package as a replacement for simp. However, function names and interval constructors are not compatible between the packages.

# In all cases, '''''y''''' contains the range of ''f'' over '''''x''''', that is, the set of ''f''('''''x''''') at points of '''''x''''' where it is defined: '''''y''''' ⊇ Rge(''f'' | '''''x''''') = {''f''(''x'') | ''x'' ∈ '''''x''''' ∩ Dom(''f'') }

−

# If also each library operation in ''f'' is everywhere defined on its inputs, while evaluating '''''y''''', then ''f'' is everywhere defined on '''''x''''', that is Dom(''f'') ⊇ '''''x'''''.

−

# If in addition, each library operation in ''f'' is everywhere continuous on its inputs, while evaluating '''''y''''', then ''f'' is everywhere continuous on '''''x'''''.

−

# If some library operation in ''f'' is nowhere defined on its inputs, while evaluating '''''y''''', then ''f'' is nowhere defined on '''''x''''', that is Dom(''f'') ∩ '''''x''''' = Ø.

−

== Quick start introduction ==

+

=== INTLAB ===

+

This interval package is ''not'' meant to be a replacement for INTLAB and any compatibility with it is pure coincidence. Since both are compatible with GNU Octave, they happen to agree on many function names and programs written for INTLAB may possibly run with this interval package as well. Some fundamental differences that I am currently aware of:

+

* INTLAB is non-free software, it grants none of the [http://www.gnu.org/philosophy/free-sw.html four essential freedoms] of free software

+

* INTLAB is not conforming to IEEE Std 1788-2015 and the parsing of intervals from strings uses a different format—especially for the uncertain form

* INTLAB uses inferior accuracy for most arithmetic operations, because it focuses on speed

+

* Basic operations can be found in both packages, but the availability of special functions depends

−

=== Input and output ===

−

Before exercising interval arithmetic, interval objects must be created, typically from non-interval data. There are interval constants <code>empty</code> and <code>entire</code> and the class constructors <code>infsup</code> for bare intervals and <code>infsupdec</code> for decorated intervals. The class constructors are very sophisticated and can be used with several kinds of parameters: Interval boundaries can be given by numeric values or string values with decimal numbers. Also it is possible to use so called interval literals with square brackets.

−

octave:1> infsup (1)

+

<div style="display:flex; align-items: flex-start">

−

ans = [1]

+

<div style="margin-right: 2em">

−

octave:2> infsup (1, 2)

+

{{Code|Computation with this interval package|<syntaxhighlight lang="octave">

−

ans = [1, 2]

+

pkg load interval

−

octave:3> infsup ("3", "4")

+

A1 = infsup (2, 3);

−

ans = [3, 4]

+

B1 = hull (-4, A2);

−

octave:4> infsup ("1.1")

+

C1 = midrad (0, 2);

−

ans = [1.0999999999999998, 1.1000000000000001]

−

octave:5> infsup ("[5, 6.5]")

−

ans = [5, 6.5]

−

octave:6> infsup ("[5.8e-17]")

−

ans = [5.799999999999999e-17, 5.800000000000001e-17]

−

It is possible to access the exact numeric interval boundaries with the functions <code>inf</code> and <code>sup</code>. The default text representation of intervals can be created with <code>intervaltotext</code>. The default text representation is not guaranteed to be exact (see function <code>intervaltoexact</code>), because this would massively spam console output. For example, the exact text representation of <code>realmin</code> would be over 700 decimal places long! However, the default text representation is correct as it guarantees to contain the actual boundaries and is accurate enough to separate different boundaries.

+

A1 + B1 * C1

+

</syntaxhighlight>

+

}}

+

</div><div>

+

{{Code|Computation with INTLAB|<syntaxhighlight lang="octave">

+

startintlab

+

A2 = infsup (2, 3);

+

B2 = hull (-4, A2);

+

C2 = midrad (0, 2);

−

octave:7> infsup (1, 1 + eps)

+

A2 + B2 * C2

−

ans = [1, 1.0000000000000003]

+

</syntaxhighlight>

−

octave:8> infsup (1, 1 + 2 * eps)

+

}}

−

ans = [1, 1.0000000000000005]

+

</div>

+

</div>

−

Warning: Decimal fractions should always be passed as a string to the constructor. Otherwise it is possible, that GNU Octave introduces conversion errors when the numeric literal is converted into floating-point format '''before''' it is passed to the constructor.

+

==== Known differences ====

−

+

Simple programs written for INTLAB should run without modification with this interval package. The following table lists common functions that use a different name in INTLAB.

−

octave:9> infsup (<span style = "color:red">0.2</span>)

+

{|

−

ans = [.20000000000000001, .20000000000000002]

+

! interval package

−

octave:10> infsup (<span style = "color:green">"0.2"</span>)

+

! INTLAB

−

ans = [.19999999999999998, .20000000000000002]

+

|-

−

+

| infsup (x)

−

For convenience it is possible to implicitly call the interval constructor during all interval operations if at least one input already is an interval object.

+

| intval (x)

−

+

|-

−

octave:11> infsup ("17.7") + 1

+

| wid (x)

−

ans = [18.699999999999999, 18.700000000000003]

+

| diam (x)

−

octave:12> ans + "[0, 2]"

+

|-

−

ans = [18.699999999999999, 20.700000000000003]

+

| subset (a, b)

−

+

| in (a, b)

−

=== Decorations ===

+

|-

−

With the subclass <code>infsupdec</code> it is possible to extend interval arithmetic with a decoration system. Every interval and intermediate result will additionally carry a decoration, which may provide additional information about the final result. The following decorations are available:

+

| interior (a, b)

−

+

| in0 (a, b)

−

{| class="wikitable" style="margin: auto"

+

|-

−

!Decoration

+

| isempty (x)

−

!Bounded

+

| isnan (x)

−

!Continuous

−

!Defined

−

!Definition

|-

|-

−

| com<br/>(common)

+

| disjoint (a, b)

−

| ✓

+

| emptyintersect (a, b)

−

| ✓

−

| ✓

−

| '''''x''''' is a bounded, nonempty subset of Dom(''f''); ''f'' is continuous at each point of '''''x'''''; and the computed interval ''f''('''''x''''') is bounded

|-

|-

−

| dac<br/>(defined &amp; continuous)

+

| hdist (a, b)

−

|

+

| qdist (a, b)

−

| ✓

−

| ✓

−

| '''''x''''' is a nonempty subset of Dom(''f''); and the restriction of ''f'' to '''''x''''' is continuous

|-

|-

−

| def<br/>(defined)

+

| disp (x)

−

|

+

| disp2str (x)

−

|

−

| ✓

−

| '''''x''''' is a nonempty subset of Dom(''f'')

|-

|-

−

| trv<br/>(trivial)

+

| infsup (s)

−

|

+

| str2intval (s)

−

|

−

|

−

| always true (so gives no information)

|-

|-

−

| ill<br/>(ill-formed)

+

| isa (x, "infsup")

−

|

+

| isintval (x)

−

|

−

|

−

| Not an interval, at least one interval constructor failed during the course of computation

|}

|}

−

In the following example, all decoration information is lost when the interval is possibly divided by zero, i. e., the overall function is not guaranteed to be defined for all possible inputs.

Arithmetic functions in a set-based interval arithmetic follow these rules: Intervals are sets. They are subsets of the set of real numbers. The interval version of an elementary function such as sin(''x'') is essentially the natural extension to sets of the corresponding point-wise function on real numbers. That is, the functions are evaluated for each number in the interval where the function is defined and the result must be an enclosure of all possible values that may occur.

One operation that should be noted is the <code>fma</code> function (fused multiply and add). It computes '''''x''''' * '''''y''''' + '''''z''''' in a single step and is much slower than multiplication followed by addition. However, it is more accurate and therefore preferred in some situations.

+

=== Architecture ===

−

octave:1> sin (infsup (0.5))

+

In a nutshell the package provides two new data types to users: bare intervals and decorated intervals. The data types are implemented as:

Almost all functions in the package are implemented as methods of these classes, e. g. <code>@infsup/sin</code> implements the sine function for bare intervals. Most code is kept in m-files. Arithmetic operations that require correctly-rounded results are implemented in oct-files (C++ code), these are used internally by the m-files of the package. The source code is organized as follows:

−

Some arithmetic functions also provide reverse mode operations. That is inverse functions with interval constraints. For example the <code>sqrrev</code> can compute the inverse of the <code>sqr</code> function on intervals.

* All methods must check <code>nargin</code> and call <code>print_usage</code> if the number of parameters is wrong. This prevents simple errors by the user.

+

* Methods with more than 1 parameter must convert non-interval parameters to intervals using the class constructor. This allows the user to mix non-interval parameters with interval parameters and the function treats any inputs as intervals. Invalid values will be handled by the class constructors.

+

if (not (isa (x, "infsup")))

+

x = infsup (x);

+

endif

+

if (not (isa (y, "infsup")))

+

y = infsup (y);

+

endif

+

+

if (not (isa (x, "infsupdec")))

+

x = infsupdec (x);

+

endif

+

if (not (isa (y, "infsupdec")))

+

y = infsupdec (y);

+

endif

+

+

==== Use of Octave functions ====

+

Octave functions may be used as long as they don't introduce arithmetic errors. For example, the ceil function can be used safely since it is exact on binary64 numbers.

+

function x = ceil (x)

+

... parameter checking ...

+

x.inf = ceil (x.inf);

+

x.sup = ceil (x.sup);

+

endfunction

+

+

If Octave functions would introduce arithmetic/rounding errors, there are interfaces to MPFR (<code>mpfr_function_d</code>) and crlibm (<code>crlibm_function</code>), which can produce guaranteed boundaries.

+

+

==== Vectorization & Indexing ====

+

All functions should be implemented using vectorization and indexing. This is very important for performance on large data. For example, consider the plus function. It computes lower and upper boundaries of the result (x.inf, y.inf, x.sup, y.sup may be vectors or matrices) and then uses an indexing expression to adjust values where empty intervals would have produces problematic values.

+

function x = plus (x, y)

+

... parameter checking ...

+

l = mpfr_function_d ('plus', -inf, x.inf, y.inf);

+

u = mpfr_function_d ('plus', +inf, x.sup, y.sup);

+

+

emptyresult = isempty (x) | isempty (y);

+

l(emptyresult) = inf;

+

u(emptyresult) = -inf;

+

…

+

endfunction

+

+

== VERSOFT ==

+

The [http://uivtx.cs.cas.cz/~rohn/matlab/ VERSOFT] software package (by Jiří Rohn) has been released under a free software license (Expat license) and algorithms may be migrated into the interval package.

| use <code>glpk</code> as a replacement for <code>linprog</code>; dependency <code>verifylss</code> is implemented as <code>mldivide</code>

+

|-

+

|verlinprogg

+

|style="color:red"| trapped

+

| depends on <code>verfullcolrank</code>

+

|-

+

|verquadprog

+

| unknown

+

| use <code>quadprog</code> from the optim package; use <code>glpk</code> as a replacement for <code>linprog</code>; dependency <code>verifylss</code> is implemented as <code>mldivide</code>; depends on <code>isspd</code> (by Rump, to be checked, algorithm in [http://www.ti3.tuhh.de/paper/rump/Ru06c.pdf])

+

|-

+

|colspan="3"|Real (or complex) data only: Polynomials

+

|-

+

|verroots

+

|style="color:red"| trapped

+

| main part implemented in <code>vereig</code>

+

|-

+

|colspan="3"|Interval (or real) data: Matrices

+

|-

+

|verhurwstab

+

|style="color:red"| trapped

+

| depends on <code style="color:red">verposdef</code>

+

|-

+

|verinverse

+

|style="color:green"| free

+

| depends on <code style="color:green">verintervalhull</code>, to be migrated

+

|-

+

|<s>verinvnonneg</s>

+

|style="color:green"| free, migrated

+

|-

+

|verposdef

+

|style="color:red"| trapped

+

| depends on <code>isspd</code> (by Rump, to be checked) and <code style="color:red">verregsing</code>

+

|-

+

|verregsing

+

|style="color:red"| trapped

+

| dependency <code>verifylss</code> is implemented as <code>mldivide</code>; depends on <code>isspd</code> (by Rump, to be checked) and <code>verintervalhull</code>; see also [http://uivtx.cs.cas.cz/~rohn/publist/singreg.pdf]

| main part implemented in <code>vereig</code>, depends on <code style="color:red">verspectrad</code>

+

|-

+

|vereigval

+

|style="color:red"| trapped

+

| depends on <code style="color:red">verregsing</code>

+

|-

+

|<s>vereigvec</s>

+

|style="color:green"| free, migrated

+

|-

+

|verperrvec

+

|style="color:green"| free

+

| the function is just a wrapper around <code style="color:green">vereigvec</code>?!?

+

|-

+

|versingval

+

|style="color:red"| trapped

+

| depends on <code style="color:red">vereigsym</code>

+

|-

+

|colspan="3"|Interval (or real) data: Matrices: Decompositions

+

|-

+

|verqr (experimental)

+

|style="color:green"| free

+

| <code>qr</code> has already been implemented using the Gram-Schmidt process, which seems to be more accurate and faster than the Cholsky decomposition or Householder reflections used in verqr. No migration needed.

+

|-

+

|<s>verchol (experimental)</s>

+

|style="color:green"| free, migrated

+

| migrated version has been named after the standard Octave function <code>chol</code>

+

|-

+

|colspan="3"|Interval (or real) data: Linear systems (square)

+

|-

+

|verenclinthull

+

|style="color:green"| free

+

| to be migrated

+

|-

+

|verhullparam

+

|style="color:green"| free

+

| depends on <code>verintervalhull</code>, to be migrated

+

|-

+

|verhullpatt

+

|style="color:green"| free

+

| depends on <code>verhullparam</code>, to be migrated

+

|-

+

|verintervalhull

+

|style="color:green"| free

+

| to be migrated

+

|-

+

|colspan="3"|Interval (or real) data: Linear systems (rectangular)

+

|-

+

|verintlinineqs

+

|style="color:green"| free

+

| depends on <code style="color:green">verlinineqnn</code>

+

|-

+

|veroettprag

+

|style="color:green"| free

+

|-

+

|vertolsol

+

|style="color:green"| free

+

| depends on <code style="color:green">verlinineqnn</code>

+

|-

+

|colspan="3"|Interval (or real) data: Matrix equations (rectangular)

+

|-

+

|vermatreqn

+

|style="color:green"| free

+

|-

+

|colspan="3"|Real data only: Uncommon problems

+

|-

+

| plusminusoneset

+

|style="color:green"| free

+

|-

+

| verabsvaleqn

+

|style="color:green"| free

+

| to be migrated

+

|-

+

| verabsvaleqnall

+

|style="color:green"| free

+

| depends on <code>verabsvaleqn</code>, see also [http://uivtx.cs.cas.cz/~rohn/publist/absvaleqnall.pdf], to be migrated

+

|-

+

| verbasintnpprob

+

|style="color:red"| trapped

+

| depends on <code style="color:red">verregsing</code>

+

|-

+

|}

−

=== Boolean operations ===

−

[[Category:Octave-Forge]]

+

[[Category:Octave Forge]]

Revision as of 04:15, 10 June 2019

Intervals are closed, connected subsets of the real numbers. Intervals may be unbound (in either or both directions) or empty. In special cases +inf and -inf are used to denote boundaries of unbound intervals, but any member of the interval is a finite real number.

Classical functions are extended to interval functions as follows: The result of function f evaluated on interval x is an interval enclosure of all possible values of f over x where the function is defined. Most interval arithmetic functions in this package manage to produce a very accurate such enclosure.

The result of an interval arithmetic function is an interval in general. It might happen, that the mathematical range of a function consist of several intervals, but their union will be returned, e. g., 1 / [-1, 1] = [Entire].

Development status

Completeness

All required functions from IEEE Std 1788-2015, IEEE standard for interval arithmetic, are implemented. The standard was approved by IEEE-SA on June 11, 2015. It will remain active for ten years. The standard was approved by ANSI in 2016.

Also, the minimalistic standard IEEE Std 1788.1-2017, IEEE standard for interval arithmetic (simplified) is fully implemented. The standard was approved by IEEE-SA on December 6, 2017 (and published in January 2018).

In addition there are functions for interval matrix arithmetic, N-dimensional interval arrays, plotting, and solvers.

a) It should be possible to specify the preferred overall field width (the length of s).
b) It should be possible to specify how Empty, Entire and NaI are output,
e.g., whether lower or upper case, and whether Entire becomes [Entire] or [-Inf, Inf].
c) For l and u, it should be possible to specify the field width,
and the number of digits after the point or the number of significant digits.
(partly this is already implemented by output_precision (...) / `format long` / `format short`)
d) It should be possible to output the bounds of an interval without punctuation,
e.g., 1.234 2.345 instead of [1.234, 2.345]. For instance, this might be a
convenient way to write intervals to a file for use by another application.

Compatibility

The interval package's main goal is to be compliant with IEEE Std 1788-2015, so it is compatible with other standard-conforming implementations (on the set of operations described by the standard document). Other implementations, which are known to aim for standard conformance are:

Octave Forge simp package

In 2008/2009 there was a Single Interval Mathematics Package (SIMP) for Octave, which has eventually become unmaintained at Octave Forge.

The simp package contains a few basic interval arithmetic operations on scalar or vector intervals. It does not consider inaccurate built-in arithmetic functions, round-off, conversion and representational errors. As a result its syntax is very easy, but the arithmetic fails to produce guaranteed enclosures.

It is recommended to use the interval package as a replacement for simp. However, function names and interval constructors are not compatible between the packages.

INTLAB

This interval package is not meant to be a replacement for INTLAB and any compatibility with it is pure coincidence. Since both are compatible with GNU Octave, they happen to agree on many function names and programs written for INTLAB may possibly run with this interval package as well. Some fundamental differences that I am currently aware of:

Almost all functions in the package are implemented as methods of these classes, e. g. @infsup/sin implements the sine function for bare intervals. Most code is kept in m-files. Arithmetic operations that require correctly-rounded results are implemented in oct-files (C++ code), these are used internally by the m-files of the package. The source code is organized as follows:

Best practices

Parameter checking

All methods must check nargin and call print_usage if the number of parameters is wrong. This prevents simple errors by the user.

Methods with more than 1 parameter must convert non-interval parameters to intervals using the class constructor. This allows the user to mix non-interval parameters with interval parameters and the function treats any inputs as intervals. Invalid values will be handled by the class constructors.

If Octave functions would introduce arithmetic/rounding errors, there are interfaces to MPFR (mpfr_function_d) and crlibm (crlibm_function), which can produce guaranteed boundaries.

Vectorization & Indexing

All functions should be implemented using vectorization and indexing. This is very important for performance on large data. For example, consider the plus function. It computes lower and upper boundaries of the result (x.inf, y.inf, x.sup, y.sup may be vectors or matrices) and then uses an indexing expression to adjust values where empty intervals would have produces problematic values.